1. POLYNOMIALS, EQUATIONS & FUNCTIONS

The roots of the equation x^2 + px + q = 0 are 1 and 2. The roots of the equation qx^2 - px + 1 = 0 must be:

-1/2 , -1

In the equation Ax^2 + 14x + 12 = 0, one root is 6 times the other root. What is the value of A?

2

Two CERTC students were asked to solve a quadratic equation, ax^2 + bx + c = 0, where “a” is not equal to 0. One of them made a mistake in writing “b” and found the roots as 3 and - 1/2 The other also made a mistake in writing c and found the roots as -1 and - 1/4. The correct roots of the given equation should be:

-2, 3/4

In the equation x^2 + 11x + 28 = 0, one root is 3 more than the other. What are the roots of the equation?

-7, -4

Determine the value of “k” so that the sum and product of roots are equal from the given equation 7x^2 + (2k - 1)x - 3k + 2 = 0.

1

The equation of whose roots are the reciprocal of the roots of 2x^2 - 3x - 5 is?

5x² + 3x - 2 = 0

Find the value of “h” in the equation 2x^2 - hx^2 + 4x + 5h = 0 so that the sum of the roots is 2.

4

Find the value of the constant “h” in the quadratic equation 3x^2 - hx + x - 7h = 0 if 3 is one of the roots.

3

Find the value of the constant "h" in the quadratic equation 2hx^2 + 3x^2 + 4x - 5h = 0 so that the product of the roots is -4.

-4

Find the sum (s) and product (p) of the roots of the equation: 3x^3 - 18x^2 + 33x - 18 = 0

6, 6

If the sum of the roots of a quadratic equation is 12 and the product is 36, then what is difference of the roots?

0

Given the equation 2x^2 + x - 10 = 0 . Find the product of roots.

-5

If (x - 1) and (x + 2) , are the factors of x³ + hx² +kx + 6 , the value of h and k are:

-2, -5

If 6x - 5y + 2 = 0 and 5x - 6y + 9 = 0 the value of x² + y² is

14.) ???

The L.C.M. of x² - 9x + 20 , x² - 6x + 5 and x³ - 5x² - x + 5 is

(x - 5)(x -4)(x - 1)(x + 1)

If a and b are the roots of the equation 4x² - 13x + 7 = 0, the value of (a + b)/ab is

13/7

If the product of the roots of 4x² + (k + 2)x - (k + 13) = 0 is - 15/4 find the sum of the roots.

-1

What is the discriminant of the equation 6x^2 + 7x - 13 = 0?

361

If P(x) = ax³ +bx² + cx +d and (b + d) - (a + c) = 0 then which following statements/expression are correct:

1. P(- 1) = 0
2. P(1) = 0
3. (x - 1) a factor p(x)
4. (x + 1) is a factor p(x)

If one root of the equation ax²+bx+c=0, with “a” not equal to 0, is reciprocal of the other root, then:

b = c

How many positive real solutions are there in the polynomial x^4 - 4x^3 + 7x^2 - 6x - 18 = 0

Determine the domain of the function:
y = SQRT(4- 3x)

Determine the domain of the function
y = 5 + SQRT(9 - x²)

Let f: R→R be defined by f(x) = 3x - 4 the f ^-1(x) is?

If f(x) = 3x and g(x) = 7x - 1, what is (f ∘ g) (4)?

If f(x) = kx + 1 and g(x) = 3x + 2 and f ∘ g=g ∘ f , then the value of k is:

If f(x) = 2x + 3 and g(x) = - x ^ 2 + 5 find f[g(x)]

Find the domain of the function f(x) = ln(x + 5)

If a^(m - 5) = 2^(2m + 10), then using the law of indices, the value of “a” is:

If x - y = 1 and x ^ 2 + y ^ 2 = 41 then the value of x + y will be:

If (2x + 3) and (x - 1) are two factors of (2x ^ 2 + x) ^ 2 - 4(2x ^ 2 + x) + 3, then the remaining factors are:

If a = b^x , b = c^y , and c = a^z then the value of xyz is equal to:

Given 32 = 70 (1-e^-t/2), determine the value of t, correct to 3 significant figures.

If 4 = sqrt{x+ sqrt[x+ sqrt(x+... ∞)]} then the value of x will be

If 4^(t - r) = 1 and 4^(x - y) = 4 then the value of x and y will be respectively

Find an interval that contains all the real zeros of P(x) = 2x ^ 3 - 5x ^ 2 + 6

a^x = b^y = c^z and b² = ac then 1/x + 1/z will be equal to

Find the value of x which will satisfy the following expression sqrt(x - 2) = sqrt(x) + 2

If x : y : z = 4 : -3 : 2 and 2x + 4y - 3z = 20, find x. y. z

The value of _____ is equal to

The value of x for which the polynomials (x ^ 2 - 1) and x ^ 2 - 2x + 1 vanish simultaneously, is

If a^2 = by + cz, b^2 = cz + ax, c^2 = ax + by then the value of x/(a + x) + y/(b + y) + z/(c + z) will be:

(x + 2) is a factor of:

A. x ^ 4 - 4x ^ 4 - 3x ^ 3 + 8x ^ 2 - 14x + 12
B. x ^ 3 + 4x ^ 4 - 3x ^ 3 + 8x ^ 2 - 14x + 12
C. x ^ 2 - 4x ^ 2 + 3x ^ 2 + 8x ^ 2 - 14x + 12
D. x ^ 3 - 4x ^ 2 - 3x ^ 3 + 8x ^ 2 + 14x + 12

The system of equations x + 2y = 3 and 2x + 4y = 3, has:

A. A unique solution
B. Infinitely many solutions
C. Has no solution
D. Has exactly two solutions

If 2x + ky + z a factor of 9y^2 - z^2 - 2xz + 6xy then the value of k is equal to:

Find the upper bounds for the real roots of the equation 2x³ - 5x² - 7x + 4 = 0

If a, b, c are the roots of the cubic equation (x - 1)(x² + x + 3) = 0 , then the values of a³ + b² + c³ are:

If 3^x - 3^(x-1) = 18 , then the value of x is

lf x^(1/3) + y^ 1/3) + z^(1/3) = 0 then the value of (x + y + z)^3 will be:

Find the value of k in the quadratic equation 3x^2 - kx + x - 7k = 0 if 3 is one of the roots.

Find the value of c from the given quadratic equation x² - 4x + c = 0 if the product of the roots is -5.

Given the equation 2x² + x - 10 = 0 Find the product of roots

Simplify (x³ + y³)/(x + y)

Divide 3x³ + x² + 3x + 5 by x+1.

What is the quadratic equation whose roots are 2/3 and - 3/4?

The roots of the quadratic equation Ax² + Bx + C = 0 are 1/4 and - 7/2 Determine the value of B.

If a^(x) b = b^(y) c = c^(z) a = 1, then (xyz)³ is equal to:

If one root of a * x ^ 2 + bx + c = 0 is double the other, then a,b,c are related as:

A. 2a ^ 2 = 9bc
B. 9a ^ 2 = 2bc
C. 2b ^ 2 = 9ac
D. 9b ^ 2 = 2ac

If A * x ^ 3 + 31x ^ 2 - Bx - 10 is exactly divisible by 2x ^ 2 + 9x - 5 then the values of A and B, respectively are:

Let p, q be non-zero integers. Consider this polynomial A(x) = x² + px + q . It is given that x - m and x - km (where m is a non-zero integer and k is a positive integer, k >= 2 ) are simple factors of A(x) which one of the following is correct?

A. (k + 1)²p² = kq
B. (k + 1)²q = kp²
C. k²q = (k + 1)p²
D. k²p² = (k + 1)²q

If x² + ax + b = 0 and x² + bx + a = 0 have a common root, then
A. a + b = 0
B. a + b = 1
C. a - b - 1 = 0
D. a + b + 1 = 0

If f(x) = 10^x + 1 find f(x + 1) - f(x)

If f(x - 1) = 1 + x² then what is f(x)

What is the remainder when 3x³ + 3x² - 4x - 13 is divided by (x - 2)?

The roots of the quadratic equation. Ax² + Bx + C = 0 are 1/4 and -7/2. Determine the value of B.

The roots of the quadratic equation Ax² + Bx + C = 0 are 3 and 2. Determine the sum of A + B + C

If one root of 9x² - 6x + k = 0 exceeds the other by 2, find the value of k.

Determine the k so that the equation 4x² + kx + 1 = 0 will have just one real solution

Find the value of k in the quadratic equation 3x² - kx + x - 7k = 0 if 3 is one of the roots.

Find the value of c from the given quadratic equation x ^ 2 - 4x + c = 0 if the product of the roots is -5.

What is the discriminant of the equation 6x² + 7x - 13 = 0

Which of the following equations has roots, which are reciprocals of the roots of the equation 2x^2 - 3x - 5 + 0
A. 3x^2 - 2x + 5 = 0
B. 2x^2 - 5x - 12 = 0
C. 7x^2 + 3x + 12 = 0
D. 5x^2 + 3x - 2 = 0

If one root of 8x^2 - 6x + k = 0 exceeds the other by 2, find the value of k

Solve |x - 3| < 4

What is the equation of the product and sum of its roots -32 and -4

The roots of a quadratic equation are: 1/3 and 1/4. What is the equation?
A. 2x ^ 2 - 17x + 10 = 0
B. x ^ 2 - 14x + 9 = 0
C. 10x ^ 2 - 5x + 2 = 0
D. 12x ^ 2 - 7x + 1 = 0

If the product of the roots of 4x² + (k + 2)x - (k + 13) = 0 is - 15/4, find the sum of the roots.

What is the quadratic equation whose roots are 2/3 and - 3/4?

If the sum of the roots of a quadratic equation is 12 and the product is 36, then what is difference of the roots?

Which of the following equations has roots which are reciprocals of the roots of the equation 2x^2 - 3x - 5 = 0?

A. 3x ^ 2 - 2x + 5 = 0
B. 2x ^ 2 - 5x - 12 = 0
C. 7x ^ 2 + 3x + 12 = 0
D. 5x ^ 2 + 3x - 2 = 0

What is the quadratic equation if the product and sum its roots -32 and 4, respectively?